Extended reals

In mathematics, the affinely extended real number system is obtained from the real number system ℝ by adding two elements: + ∞ and – ∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted ${\displaystyle {\overline {\mathbb {R} }}}$ or [–∞, +∞] or ℝ ∪ {–∞, +∞}.

When the meaning is clear from context, the symbol +∞ is often written simply as ∞.

We often wish to describe the behavior of a function ${\displaystyle f(x)}$, as either the argument ${\displaystyle x}$ or the function value ${\displaystyle f(x)}$ gets "very big" in some sense. For example, consider the function

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